TY - JOUR

T1 - Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons

AU - Krivulin, Nikolai

N1 - Krivulin N. Algebraic solution to box-constrained bi-criteria problem of rating alternatives through pairwise comparisons // Kybernetika. 2022. Vol. 58, N 5. P. 665-690. DOI: 10.14736/kyb-2022-5-0665

PY - 2022

Y1 - 2022

N2 - We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then represent the problem in terms of tropical algebra. We apply methods and results of tropical optimization to derive an analytical solution of the constrained problem. The solution consists in introducing two new variables that describe the values of the objective functions and allow reducing the problem to the solution of a system of parameterized inequalities constructed for the unknown vector, where the new variables play the role of parameters. We exploit the existence condition for solutions of the system to derive those values of the parameters that belong to the Pareto front inherent to the problem. Then, we solve the system for the unknown vector and take all solutions that correspond to the Pareto front, as a complete solution of the bi-objective problem. We apply the result obtained to the bi-criteria decision problem under consideration and present illustrative examples.

AB - We consider a decision-making problem to evaluate absolute ratings of alternatives that are compared in pairs according to two criteria, subject to box constraints on the ratings. The problem is formulated as the log-Chebyshev approximation of two pairwise comparison matrices by a common consistent matrix (a symmetrically reciprocal matrix of unit rank), to minimize the approximation errors for both matrices simultaneously. We rearrange the approximation problem as a constrained bi-objective optimization problem of finding a vector that determines the approximating consistent matrix, and then represent the problem in terms of tropical algebra. We apply methods and results of tropical optimization to derive an analytical solution of the constrained problem. The solution consists in introducing two new variables that describe the values of the objective functions and allow reducing the problem to the solution of a system of parameterized inequalities constructed for the unknown vector, where the new variables play the role of parameters. We exploit the existence condition for solutions of the system to derive those values of the parameters that belong to the Pareto front inherent to the problem. Then, we solve the system for the unknown vector and take all solutions that correspond to the Pareto front, as a complete solution of the bi-objective problem. We apply the result obtained to the bi-criteria decision problem under consideration and present illustrative examples.

KW - idempotent semifield

KW - tropical optimization

KW - Constrained bi-criteria decision problem

KW - Pareto-optimal solution

KW - box constraints

KW - pairwise comparisons

U2 - 10.14736/kyb-2022-5-0665

DO - 10.14736/kyb-2022-5-0665

M3 - Article

VL - 58

SP - 665

EP - 690

JO - Kybernetika

JF - Kybernetika

SN - 0023-5954

IS - 5

ER -